BZ and the Turing Instability
Tamas Bansagi
BZ Boot camp @ Brandeis
What are these two patterns?
Tropical fish
Turing patterns in
a chemical reaction
Alan Turing’s theory
‘The chemical basis of morphogenesis’
Philosophical Transactions of the Royal Society of London, (Series B, No.641, Vol. 237, 37-72,1952).
Consider:
du
 f (u , v)  Du  2u
dt
dv
 g (u, v)  Dv  2 v
dt
Kinetics+
u activator
Nonlinearity
v inhibitor
Di diffusion coefficients
f , g kinetic rate equations
Diffusive
Transport
Reaction-Diffusion equations
- In a Reaction-Diffusion system, patterns stationary in time and periodic in space
may develop if Du≠Dv.
- In the same system, if Du=Dv≥0 u and v tend to a stable uniform steady state.
- More precisely:
Du<Dv
(Long range inhibition, short range activation required)
Alan Turing’s theory
Morphogenesis (development of pattern and form)
Chemical pre-patterning through diffusion driven instability.
Positional information template
Cell differentiation, migration, shape change
Formation and development of embryo
Early stage
Turing patterns in experiment
Living systems:
• Difficult to identify pre-patterning species (morphogens)
• Mechanisms are very complicated
Chemical systems:
• Relatively easy to identify species
• Mechanisms tend to be simpler
• Seemed easier to find/design systems supporting Turing patterns
Turing patterns in experiment
Living systems:
• Difficult to identify pre-patterning species (morphogens)
• Mechanisms are very complicated
Chemical systems:
• Relatively easy to identify species
• Mechanisms tend to be much simpler
• Seemed easier to find/design systems supporting Turing patterns
• Reality: first Turing patterns reported in 1990 – Clorite-Iodide-Malonic acid reaction
(V. Castets, E. Dulos, J. Boissonade, P. De Kepper, 1990)
Examples from Biology:
• Disposition of feather buds in chick (H. S. Jung, 1998)
• Hair follicles in mice (S. Sick, S. Reinker, J. Timmer, T. Schlake, 2006)
• Skin pattern regeneration in zebra fish (M. Yamaguchi, E. Yoshimoto, S. Kondo,
2007)
Turing patterns in the BZ reaction
Oregonator model
X
Y
Z
Oregonator model in dimensionless form
activator
inhibitor
oxidized form of catalyst
Turing patterns in the BZ reaction
1D Oregonator reaction-diffusion system
Dx=Dy=Dz=1
Turing patterns in the BZ reaction
1D Oregonator reaction-diffusion system – Homogeneous perturbation
Dx=0.01, Dy=Dz=1
Turing patterns in the BZ reaction
1D Oregonator reaction-diffusion system – Inhomogeneous perturbation
Dx=0.01, Dy=Dz=1
It is in the model but how can we “slow down” the activator or “speed up” the inhibitor?
Turing patterns – BZ-AOT system
Water-in-oil microemulsion
Rh = 5-20 nm
Communication between droplets
• collision (fusion and fission) ~ 10-3 s time scale
(exchange of polar species)
• nonpolar species in oil ~ 10-4 - 10-5 s time scale
Role of Br2
• produced in the reaction
• quickly diffuses in the oil phase
• its reaction with malonic acid gives bromide (Y)
(Thorough review – V. K. Vanag and I. R. Epstein, 2008)
Oil (Octane)
Aqueous BZ chemicals
AOT – Aerosol OT sodium bis(2-ethylhexyl)
sulfosuccinate
Long range inhibition
Turing patterns – BZ-AOT system
2D
3D
Experiments (oil: cyclooctane)
Reconstructed patterns
Reconstruction
(inverse Radon transform)
Numerical results in an Oregonator-based model
(T. Bansagi, V. K. Vanag, I. R. Epstein, 2011)